How many permutations of the integers 1 through N consist of a strictly ascending sequence followed by a strictly descending sequence?

For example, for N = 9, one such permutation is (1-4-5-7-9-8-6-3-2). There must be at least two integers in a sequence, and N is considered to be a member of both sequences.

Reversals are not considered to be different permutations, i.e. (2-3-6-8-9-7-5-4-1) is the same as the above example.

(In reply to Derivation of formula by Jer)

To clear of any ambiguity, the first paragraph says that the whole number consists of 2 sequences. The second paragraph says that each sequence has at least two digits and that N, the largest digit, belongs to each of these tow sequences. So N can't be at an end.

Posted by Charlie on 2025-01-11 14:05:40